WordPress saves them in the databas in wp_usermeta as serilized array.
For the dashboard all are registered in meta-box-order_dashboard if the meta-box(s) are in metaboxhidden_dashboard they will not be shown. The order will be as saved in database and if not in metaboxhidden_dashboard.
Example the dashboard:
KEY: meta-box-order_dashboard
VALUE:
Array
(
[normal] => dashboard_right_now,dashboard_recent_comments,dashboard_incoming_links,dashboard_plugins
[side] => dashboard_quick_press,dashboard_recent_drafts,dashboard_primary,dashboard_secondary
[column3] =>
[column4] =>
)
There are also these meta_key(s):
- meta-box-order_dashboard
- metaboxhidden_dashboard
- meta-box-order_page
- metaboxhidden_page
- metaboxhidden_nav-menus
- meta-box-order_{custom_post}